Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.3% → 99.8%

Time: 7.8s

Alternatives: 11

Speedup: 0.5×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x + y \cdot \left(z - x\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y \cdot \left(z - x\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+56} \lor \neg \left(y \leq 2 \cdot 10^{+57}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e+56) (not (<= y 2e+57)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ (* x (- 1.0 y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+56) || !(y <= 2e+57)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x * (1.0 - y)) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1d+56)) .or. (.not. (y <= 2d+57))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + ((x * (1.0d0 - y)) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+56) || !(y <= 2e+57)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x * (1.0 - y)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e+56) or not (y <= 2e+57):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + ((x * (1.0 - y)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e+56) || !(y <= 2e+57))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(Float64(x * Float64(1.0 - y)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e+56) || ~((y <= 2e+57)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + ((x * (1.0 - y)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e+56], N[Not[LessEqual[y, 2e+57]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000009e56 or 2.0000000000000001e57 < y

   1. Initial program 73.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.6%

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 95.6%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) + y} \]

      2. fma-def 95.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \frac{y}{z} + \frac{1}{z}, y\right)} \]

      3. +-commutative 95.6%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} + -1 \cdot \frac{y}{z}}, y\right) \]

      4. mul-1-neg 95.6%

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{z} + \color{blue}{\left(-\frac{y}{z}\right)}, y\right) \]

      5. unsub-neg 95.6%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} - \frac{y}{z}}, y\right) \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right)} \]

   6. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{z}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto y \cdot \left(1 + \color{blue}{\left(-\frac{x}{z}\right)}\right) \]

      2. unsub-neg 99.9%

         \[\leadsto y \cdot \color{blue}{\left(1 - \frac{x}{z}\right)} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{x}{z}\right)} \]

   if -1.00000000000000009e56 < y < 2.0000000000000001e57

   1. Initial program 99.3%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{y + \frac{x \cdot \left(1 + -1 \cdot y\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+56} \lor \neg \left(y \leq 2 \cdot 10^{+57}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (- (/ 1.0 z) (/ y z)) y))

C

double code(double x, double y, double z) {
	return fma(x, ((1.0 / z) - (y / z)), y);
}

Julia

function code(x, y, z)
	return fma(x, Float64(Float64(1.0 / z) - Float64(y / z)), y)
end

Wolfram

code[x_, y_, z_] := N[(x * N[(N[(1.0 / z), $MachinePrecision] - N[(y / z), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.0%

   \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
4. Step-by-step derivation

   1. +-commutative 98.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) + y} \]

   2. fma-def 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \frac{y}{z} + \frac{1}{z}, y\right)} \]

   3. +-commutative 98.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} + -1 \cdot \frac{y}{z}}, y\right) \]

   4. mul-1-neg 98.0%

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{z} + \color{blue}{\left(-\frac{y}{z}\right)}, y\right) \]

   5. unsub-neg 98.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} - \frac{y}{z}}, y\right) \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right)} \]

6. Final simplification 98.0%

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right) \]
7. Add Preprocessing

Alternative 3: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-92}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.0152:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y z))))
   (if (<= y -1.15e-16)
     t_0
     (if (<= y 3.3e-109)
       (/ x z)
       (if (<= y 3.1e-92) y (if (<= y 0.0152) (/ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -1.15e-16) {
		tmp = t_0;
	} else if (y <= 3.3e-109) {
		tmp = x / z;
	} else if (y <= 3.1e-92) {
		tmp = y;
	} else if (y <= 0.0152) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y / z)
    if (y <= (-1.15d-16)) then
        tmp = t_0
    else if (y <= 3.3d-109) then
        tmp = x / z
    else if (y <= 3.1d-92) then
        tmp = y
    else if (y <= 0.0152d0) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -1.15e-16) {
		tmp = t_0;
	} else if (y <= 3.3e-109) {
		tmp = x / z;
	} else if (y <= 3.1e-92) {
		tmp = y;
	} else if (y <= 0.0152) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / z)
	tmp = 0
	if y <= -1.15e-16:
		tmp = t_0
	elif y <= 3.3e-109:
		tmp = x / z
	elif y <= 3.1e-92:
		tmp = y
	elif y <= 0.0152:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / z))
	tmp = 0.0
	if (y <= -1.15e-16)
		tmp = t_0;
	elseif (y <= 3.3e-109)
		tmp = Float64(x / z);
	elseif (y <= 3.1e-92)
		tmp = y;
	elseif (y <= 0.0152)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / z);
	tmp = 0.0;
	if (y <= -1.15e-16)
		tmp = t_0;
	elseif (y <= 3.3e-109)
		tmp = x / z;
	elseif (y <= 3.1e-92)
		tmp = y;
	elseif (y <= 0.0152)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-16], t$95$0, If[LessEqual[y, 3.3e-109], N[(x / z), $MachinePrecision], If[LessEqual[y, 3.1e-92], y, If[LessEqual[y, 0.0152], N[(x / z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e-16 or 0.0152 < y

   1. Initial program 76.6%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 37.9%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 33.1%

      \[\leadsto \frac{\color{blue}{y \cdot z}}{z} \]
   5. Step-by-step derivation

      1. associate-/l* 50.0%

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{z}}} \]

      2. associate-/r/ 53.8%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]

   6. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]

   if -1.15e-16 < y < 3.2999999999999999e-109 or 3.1000000000000001e-92 < y < 0.0152

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 74.0%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if 3.2999999999999999e-109 < y < 3.1000000000000001e-92

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-92}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.0152:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-17}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-109} \lor \neg \left(y \leq 3.1 \cdot 10^{-92}\right) \land y \leq 0.0135:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-17)
   y
   (if (or (<= y 3.3e-109) (and (not (<= y 3.1e-92)) (<= y 0.0135)))
     (/ x z)
     y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-17) {
		tmp = y;
	} else if ((y <= 3.3e-109) || (!(y <= 3.1e-92) && (y <= 0.0135))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-17)) then
        tmp = y
    else if ((y <= 3.3d-109) .or. (.not. (y <= 3.1d-92)) .and. (y <= 0.0135d0)) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-17) {
		tmp = y;
	} else if ((y <= 3.3e-109) || (!(y <= 3.1e-92) && (y <= 0.0135))) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-17:
		tmp = y
	elif (y <= 3.3e-109) or (not (y <= 3.1e-92) and (y <= 0.0135)):
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-17)
		tmp = y;
	elseif ((y <= 3.3e-109) || (!(y <= 3.1e-92) && (y <= 0.0135)))
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-17)
		tmp = y;
	elseif ((y <= 3.3e-109) || (~((y <= 3.1e-92)) && (y <= 0.0135)))
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-17], y, If[Or[LessEqual[y, 3.3e-109], And[N[Not[LessEqual[y, 3.1e-92]], $MachinePrecision], LessEqual[y, 0.0135]]], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000014e-17 or 3.2999999999999999e-109 < y < 3.1000000000000001e-92 or 0.0134999999999999998 < y

   1. Initial program 77.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 52.3%

      \[\leadsto \color{blue}{y} \]

   if -2.00000000000000014e-17 < y < 3.2999999999999999e-109 or 3.1000000000000001e-92 < y < 0.0134999999999999998

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 74.0%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-17}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-109} \lor \neg \left(y \leq 3.1 \cdot 10^{-92}\right) \land y \leq 0.0135:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 10^{+224}:\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+274}:\\ \;\;\;\;y \cdot \left(-\frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.0)
   (+ y (/ x z))
   (if (<= y 1e+224) (- y (/ x z)) (if (<= y 6.9e+274) (* y (- (/ x z))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		tmp = y + (x / z);
	} else if (y <= 1e+224) {
		tmp = y - (x / z);
	} else if (y <= 6.9e+274) {
		tmp = y * -(x / z);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = y + (x / z)
    else if (y <= 1d+224) then
        tmp = y - (x / z)
    else if (y <= 6.9d+274) then
        tmp = y * -(x / z)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		tmp = y + (x / z);
	} else if (y <= 1e+224) {
		tmp = y - (x / z);
	} else if (y <= 6.9e+274) {
		tmp = y * -(x / z);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.0:
		tmp = y + (x / z)
	elif y <= 1e+224:
		tmp = y - (x / z)
	elif y <= 6.9e+274:
		tmp = y * -(x / z)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 1e+224)
		tmp = Float64(y - Float64(x / z));
	elseif (y <= 6.9e+274)
		tmp = Float64(y * Float64(-Float64(x / z)));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = y + (x / z);
	elseif (y <= 1e+224)
		tmp = y - (x / z);
	elseif (y <= 6.9e+274)
		tmp = y * -(x / z);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+224], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.9e+274], N[(y * (-N[(x / z), $MachinePrecision])), $MachinePrecision], y]]]

Derivation

1. Split input into 4 regimes

2. if y < 1

   1. Initial program 93.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 85.2%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 89.8%

      \[\leadsto \color{blue}{y + \frac{x}{z}} \]

   if 1 < y < 9.9999999999999997e223

   1. Initial program 79.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 34.0%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 51.0%

      \[\leadsto \color{blue}{y + \frac{x}{z}} \]
   5. Step-by-step derivation

      1. div-inv 51.0%

         \[\leadsto y + \color{blue}{x \cdot \frac{1}{z}} \]

      2. add-sqr-sqrt 23.9%

         \[\leadsto y + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{z} \]

      3. sqrt-unprod 68.1%

         \[\leadsto y + \color{blue}{\sqrt{x \cdot x}} \cdot \frac{1}{z} \]

      4. sqr-neg 68.1%

         \[\leadsto y + \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \cdot \frac{1}{z} \]

      5. sqrt-unprod 41.1%

         \[\leadsto y + \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \frac{1}{z} \]

      6. add-sqr-sqrt 66.1%

         \[\leadsto y + \color{blue}{\left(-x\right)} \cdot \frac{1}{z} \]

      7. cancel-sign-sub-inv 66.1%

         \[\leadsto \color{blue}{y - x \cdot \frac{1}{z}} \]

      8. div-inv 66.1%

         \[\leadsto y - \color{blue}{\frac{x}{z}} \]

   6. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{y - \frac{x}{z}} \]

   if 9.9999999999999997e223 < y < 6.8999999999999997e274

   1. Initial program 73.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 73.4%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate-*l/ 86.5%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(z - x\right)} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(z - x\right)} \]

   6. Taylor expanded in z around 0 57.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*l/ 65.4%

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{z} \cdot y\right)} \]

      2. associate-*l* 65.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{z}\right) \cdot y} \]

      3. *-commutative 65.4%

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{x}{z}\right)} \]

      4. associate-*r/ 65.4%

         \[\leadsto y \cdot \color{blue}{\frac{-1 \cdot x}{z}} \]

      5. neg-mul-1 65.4%

         \[\leadsto y \cdot \frac{\color{blue}{-x}}{z} \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{y \cdot \frac{-x}{z}} \]

   if 6.8999999999999997e274 < y

   1. Initial program 39.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 10^{+224}:\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+274}:\\ \;\;\;\;y \cdot \left(-\frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+38} \lor \neg \left(y \leq 1.2 \cdot 10^{+57}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.6e+38) (not (<= y 1.2e+57)))
   (* y (- 1.0 (/ x z)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e+38) || !(y <= 1.2e+57)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.6d+38)) .or. (.not. (y <= 1.2d+57))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e+38) || !(y <= 1.2e+57)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.6e+38) or not (y <= 1.2e+57):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.6e+38) || !(y <= 1.2e+57))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.6e+38) || ~((y <= 1.2e+57)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.6e+38], N[Not[LessEqual[y, 1.2e+57]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5999999999999999e38 or 1.20000000000000002e57 < y

   1. Initial program 72.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.7%

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 95.7%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) + y} \]

      2. fma-def 95.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \frac{y}{z} + \frac{1}{z}, y\right)} \]

      3. +-commutative 95.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} + -1 \cdot \frac{y}{z}}, y\right) \]

      4. mul-1-neg 95.7%

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{z} + \color{blue}{\left(-\frac{y}{z}\right)}, y\right) \]

      5. unsub-neg 95.7%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} - \frac{y}{z}}, y\right) \]

   5. Simplified 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right)} \]

   6. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{z}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto y \cdot \left(1 + \color{blue}{\left(-\frac{x}{z}\right)}\right) \]

      2. unsub-neg 99.9%

         \[\leadsto y \cdot \color{blue}{\left(1 - \frac{x}{z}\right)} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{x}{z}\right)} \]

   if -2.5999999999999999e38 < y < 1.20000000000000002e57

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+38} \lor \neg \left(y \leq 1.2 \cdot 10^{+57}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+71} \lor \neg \left(x \leq 2.65 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.8e+71) (not (<= x 2.65e+38)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e+71) || !(x <= 2.65e+38)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.8d+71)) .or. (.not. (x <= 2.65d+38))) then
        tmp = x * ((1.0d0 - y) / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.8e+71) || !(x <= 2.65e+38)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.8e+71) or not (x <= 2.65e+38):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.8e+71) || !(x <= 2.65e+38))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.8e+71) || ~((x <= 2.65e+38)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.8e+71], N[Not[LessEqual[x, 2.65e+38]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.79999999999999978e71 or 2.65000000000000012e38 < x

   1. Initial program 87.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) + y} \]

      2. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \frac{y}{z} + \frac{1}{z}, y\right)} \]

      3. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} + -1 \cdot \frac{y}{z}}, y\right) \]

      4. mul-1-neg 99.6%

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{z} + \color{blue}{\left(-\frac{y}{z}\right)}, y\right) \]

      5. unsub-neg 99.6%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} - \frac{y}{z}}, y\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right)} \]

   6. Taylor expanded in x around inf 86.5%

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{z} - \frac{y}{z}\right)} \]
   7. Step-by-step derivation

      1. div-sub 86.5%

         \[\leadsto x \cdot \color{blue}{\frac{1 - y}{z}} \]

   8. Simplified 86.5%

      \[\leadsto \color{blue}{x \cdot \frac{1 - y}{z}} \]

   if -8.79999999999999978e71 < x < 2.65000000000000012e38

   1. Initial program 89.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 77.5%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 88.0%

      \[\leadsto \color{blue}{y + \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+71} \lor \neg \left(x \leq 2.65 \cdot 10^{+38}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+46} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.8e+46) (not (<= y 1.0)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+46) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.8d+46)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+46) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.8e+46) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.8e+46) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.8e+46) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.8e+46], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000018e46 or 1 < y

   1. Initial program 75.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 96.0%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) + y} \]

      2. fma-def 96.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \frac{y}{z} + \frac{1}{z}, y\right)} \]

      3. +-commutative 96.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} + -1 \cdot \frac{y}{z}}, y\right) \]

      4. mul-1-neg 96.0%

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{z} + \color{blue}{\left(-\frac{y}{z}\right)}, y\right) \]

      5. unsub-neg 96.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z} - \frac{y}{z}}, y\right) \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{z} - \frac{y}{z}, y\right)} \]

   6. Taylor expanded in y around inf 98.9%

      \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{z}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto y \cdot \left(1 + \color{blue}{\left(-\frac{x}{z}\right)}\right) \]

      2. unsub-neg 98.9%

         \[\leadsto y \cdot \color{blue}{\left(1 - \frac{x}{z}\right)} \]

   8. Simplified 98.9%

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{x}{z}\right)} \]

   if -2.80000000000000018e46 < y < 1

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 98.9%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{y + \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+46} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.0) (+ y (/ x z)) (- y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = y - (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 93.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 85.2%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 89.8%

      \[\leadsto \color{blue}{y + \frac{x}{z}} \]

   if 1 < y

   1. Initial program 75.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 29.7%

      \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

   4. Taylor expanded in x around 0 47.2%

      \[\leadsto \color{blue}{y + \frac{x}{z}} \]
   5. Step-by-step derivation

      1. div-inv 47.2%

         \[\leadsto y + \color{blue}{x \cdot \frac{1}{z}} \]

      2. add-sqr-sqrt 20.2%

         \[\leadsto y + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{z} \]

      3. sqrt-unprod 64.9%

         \[\leadsto y + \color{blue}{\sqrt{x \cdot x}} \cdot \frac{1}{z} \]

      4. sqr-neg 64.9%

         \[\leadsto y + \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \cdot \frac{1}{z} \]

      5. sqrt-unprod 38.7%

         \[\leadsto y + \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \frac{1}{z} \]

      6. add-sqr-sqrt 59.6%

         \[\leadsto y + \color{blue}{\left(-x\right)} \cdot \frac{1}{z} \]

      7. cancel-sign-sub-inv 59.6%

         \[\leadsto \color{blue}{y - x \cdot \frac{1}{z}} \]

      8. div-inv 59.6%

         \[\leadsto y - \color{blue}{\frac{x}{z}} \]

   6. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{y - \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ y + \frac{x}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ y (/ x z)))

C

double code(double x, double y, double z) {
	return y + (x / z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y + (x / z)
end function

Java

public static double code(double x, double y, double z) {
	return y + (x / z);
}

Python

def code(x, y, z):
	return y + (x / z)

Julia

function code(x, y, z)
	return Float64(y + Float64(x / z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = y + (x / z);
end

Wolfram

code[x_, y_, z_] := N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 69.6%

   \[\leadsto \frac{x + \color{blue}{y \cdot z}}{z} \]

4. Taylor expanded in x around 0 77.8%

   \[\leadsto \color{blue}{y + \frac{x}{z}} \]

5. Final simplification 77.8%

   \[\leadsto y + \frac{x}{z} \]
6. Add Preprocessing

Alternative 11: 40.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 y)

C

double code(double x, double y, double z) {
	return y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y
end function

Java

public static double code(double x, double y, double z) {
	return y;
}

Python

def code(x, y, z):
	return y

Julia

function code(x, y, z)
	return y
end

MATLAB

function tmp = code(x, y, z)
	tmp = y;
end

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 40.1%

   \[\leadsto \color{blue}{y} \]

4. Final simplification 40.1%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (/ x z)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y + (x / z)) - (y / (z / x))
end function

Java

public static double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / x));
}

Python

def code(x, y, z):
	return (y + (x / z)) - (y / (z / x))

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(x / z)) - Float64(y / Float64(z / x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (x / z)) - (y / (z / x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024017 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))